Squid swimming in water fluids problem

1. The problem statement, all variables and given/known data
Some underwater fish use a jet to move forward. The body expands with water and contracts, expelling water and thus propelling forward. For these purposes, assume that a submerged fish experiences a friction drag proportional to its surface area ##A_s## and proportional to the square of its speed. Assume that the drag coefficient ##C_d## is constant. The fish also experiences an additional inviscid drag during acceleration: for the fish to accelerate, it must also accelerate some of the water around it. This effect is called added-mass. Take the added mass to be ##\alpha## times the mass of water displaced by the fish. For a spherical shape, ##\alpha = 1/2##, while a more streamlined body will have a lower ##\alpha##.

At ##t = 0##, the fish is at rest at ##x = 0##. Develop the equations necessary to determine the position of the fish ##x(t)##. Both the fish tissue and water are incompressible.

2. Relevant equations
Conservation of momentum comes to mind.

3. The attempt at a solution
Before analyzing this situation, let's define some nomenclature. Let the mass of the squid (fish) without the propellent (water) be ##m_f = \rho_f V_f##, where ##V_f## is the inherent volume of the fish (muscles and bones) excluding the volume of water in its internal cavity. Let the volume of water in the fish at some time be ##V_w(t)##. When the fish uses its muscles to pressurize the body cavity and ejects a water jet through an orifice, let that area be ##A_j##. Let the velocity of the water jet relative to the fish’s body be ##V_j(t)##. Let's also adopt a 2-D cartesian plane, where the fish moves in the ##\hat{j}## direction. For ease, let's just analyze the fish as it moves forward on one out squeeze (I think this will be easier to start with rather than considering also the fish expanding and then contracting again). Let the control volume be the fish itself and the fluid in the fish's cavity. Then conservation of momentum is expressed as
$$
\frac{\partial}{\partial t}\iiint_V \rho \vec{u} \, dV =- \iint_{\partial V} \rho \vec{u} ( \vec{u}_{rel} \cdot \vec{n} )\, dS - \iint_{\partial V} P \vec{n} \, dS + \vec{F}_{drag} \implies\\
\frac{\partial}{\partial t}\iiint_V \rho_f x'(t) \hat{j}\, dV_f + \rho_w x'(t) \hat{j} \, dV_w =- \iint_{\partial V} \rho_w x'(t) \hat{j} ( -V_j \hat{j} \cdot (-\hat{j}) )\, dS - \iint_{\partial V} P \vec{n} \, dS - (C_d A_s x'(t)^2 +\alpha (V_f+V_w))\hat{j} \implies\\
m_f x''(t) \hat{j} + V_w'(t) \rho_w x'(t) \hat{j}+V_w(t) \rho_w x''(t) \hat{j} =-A_j \rho_w x'(t) V_j \hat{j} - \iint_{\partial V} P \vec{n} \, dS - (C_d A_s x'(t)^2 +\alpha (V_f+V_w))\hat{j}$$

Do you agree with what I have here? I don't really know how to proceed here. How do I deal with pressure ##P##? Also, I know the volume integral for water is a function of time; does continuity help me here?

Awesome, thanks so much! This was great confirmation that I am in fact on the right track. Referencing equation (23) in the paper you attached, it appears my final expression is very close to theirs. However, there are two differences perhaps you could help with? Below is my final derived equation:

To me, I see three big differences. Firstly, it appears Renda et. al do not consider pressure. This makes sense to me since the open orifice allows for constant pressure in and out of the fish, so I will omit the pressure term. Doing so and invoking continuity, which I forgot to do, yields for my equation
$$(m_w(t)+m_f )x''(t) =- (C_d A_s x'(t)^2 +\alpha (V_f+V_w))$$
which is very close to what Renda has. However, I have omitted the thrust force. Can you help me understand how I know to include this and how to evaluate it?